The one-way analysis of variance with ordered alternatives: A modification of bartholomew's Ē2 Test

Tests of homogeneity of normal means with the alternative restricted by an ordering on the means are considered. The simply ordered case, μ₁ ≤ μ₂ ≤ ··· ≤ μ_k, and the simple tree ordering, μ₁ ≤ μ_j, for; j= 2, 3,…, k, are emphasized. A modification of the likelihood-ratio test is proposed which is asymptotically equivalent to it but is more robust to violations of the hypothesized orderings. The new test has power at the points satisfying the hypothesized ordering which is similar to that of the likelihood-ratio test provided the degrees of freedom are not too small. The modified test is shown to be unbiased and consistent.